Koornwinder polynomials

introduction

The Koornwinder polynomials, introduced in [Koornwinder92], are a generalisation of the Macdonald polynomials to the root system BCn.
- $n = 1$, they correspond to the Askey-Wilson polynomials depending on 5-parameter $(q;t_0,t_1,t_2,t_3)$ or $(q;a,b,c,d)$
- $n \geq 2$, they depend on six parameters $(q,t;t_0,t_1,t_2,t_3)$
It is known that all Macdonald polynomials associated with the classical (nonexceptional) root systems can be considered as their special cases, see Macdonald polynomials associatied with root systems
In Macdonald's theory of orthogonal polynomials associated to root systems, the Koornwinder-Macdonald polynomials is a family which corresponds to the non-reduced irreducible affine root system of type $(C^{\vee}_n,C_n)$

definition

def (Koornwinder density)

Throughout this section $x=(x_1,\dots,x_n)$. Then the Koornwinder density is given by \begin{equation}\label{Eq_Kdensity} \Delta(x;q,t;t_0,t_1,t_2,t_3):= \prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} {\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} \prod_{1\leq i<j\leq n} \frac{(x_i^{\pm}x_j^{\pm};q)_{\infty}} {(tx_i^{\pm}x_j^{\pm};q)_{\infty}}, \end{equation} where \begin{align*} (x_i^{\pm};q)_{\infty}&:=(x_i,x_i^{-1};q)_{\infty} \\ (x_i^{\pm}x_j^{\pm};q)_{\infty}&:= (x_ix_j,x_ix_j^{-1},x_i^{-1}x_j,x_i^{-1}x_j^{-1};q)_{\infty}. \end{align*} For complex $q,t,t_0,\dots,t_3$ such that $\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1$ this defines a scalar product on $\mathbb{C}[x^{\pm 1}]$ via \[ \langle{f}, {g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;t_0,t_1,t_2,t_3) \,d T(x), \] where \[ d T(x):=\frac{1}{2^n n! (2\pi i)^n}\, \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. \] Let $W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group with natural action on $\mathbb{C}[x^{\pm}]$. For $\lambda$ a partition of length at most $n$, let $m_{\lambda}^W$ be the $W$-invariant monomial symmetric function \[ m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} \] summed over all $\alpha$ in the $W$-orbit of $\lambda$.

def (Koornwinder polynomial)

In analogy with the Macdonald polynomials, the Koornwinder polynomials $K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)$ are defined as the unique family of polynomials in $\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W$ such that [43] \[ K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} \] and \begin{equation}\label{Eq_KKnul} \langle K_{\lambda}, K_{\mu}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =0 \qquad\text{if }\lambda\neq\mu. \end{equation}

notation

$ \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\tees}{t_0,t_1,t_2,t_3} \newcommand{\B}{\mathrm B} \newcommand{\BC}{\mathrm{BC}} \newcommand{\C}{\mathrm C} \newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} $

properties

From the definition it follows that the $K_{\lambda}$ are symmetric under permutation of the $t_r$.

quadratic norm

The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) and \cite{Sahi99} (general case). For our purposes we only need \begin{equation}\label{Eq_Gus} \langle{1},{1}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =\prod_{i=1}^n \frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} {(q,t^i;q)_{\infty}\prod_{0\leq r<s\leq 3}(t_rt_st^{i-1};q)_{\infty}}, \end{equation} known as Gustafson's integral [Gustafson90]

Cauchy identity

thm (Mimachi, [Mimachi01] thm 2.1)

The $\mathrm{BC}_n$ analogue of the Cauchy identity is given by \begin{align}\label{Eq_Mim} \sum_{\lambda\subseteq m^n} (-1)^{\lvert{\lambda}\rvert} K_{m^n-\lambda}(x;q,t;t_0,t_1,t_2,t_3) K_{\lambda'}(y;t,q;t_0,t_1,t_2,t_3) \\ &=\prod_{i=1}^n\prod_{j=1}^m \big(x_i+x_i^{-1}-y_j-y_j^{-1}\big)\\ &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), \end{align} where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.

expression in terms of interpolation polynomials

binomial coefficient

$$ {\lambda \brack \mu}_{q,t,s} := \frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})} {\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})} $$

Okounkov's binomial formula (Theorem 7.10, [Okounkov98]) gives an expansion of the Koornwinder polynomials in terms of BCn interpolation polynomials
The coefficients in this expansion are given by $BC_n$ $q$-binomial coefficients ${\lambda \brack \mu}_{q,t,s}$ times a ratio of principally specialised Koornwinder polynomials:

$$ K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} {\lambda \brack \mu}_{q,t,s} \, \frac{K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)} {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), $$ where $s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}$.

Now let \cite{Rains05} \[ C_{\lambda}^{+}(z;q,t):=\prod_{(i,j)\in \lambda} \big(1-zq^{\lambda_i+j-1}t^{2-\lambda'_j-i}\big). \] By \cite[Proposition 4.1]{Rains05} \[ \qbin{m^n}{\mu}_{q,t,s}=(-q)^{\abs{\mu}} t^{n(\mu)} q^{n(\mu')}\, \frac{(t^n,q^{-m},s^2q^mt^{1-n};q,t)_{\mu}} {C_{\mu}^{-}(q,t;q,t)C_{\mu}^{+}(s^2;q,t)} \] and the specialisation formulas \cite{vDiejen96,Sahi99} \begin{multline*} K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;\tees\big) \\ =\frac{t^{n(\lambda)}}{(t_0t^{n-1})^{\abs{\la}}} \cdot \frac{(t^n,t_0t_1t^{n-1},t_0t_2t^{n-1},t_0t_3t^{n-1};q,t)_{\la}} {C^{-}_{\la}(t;q,t)C^{+}_{\la}(t_0t_1t_2t_3t^{2n-2}/q;q,t)} \end{multline*} and \cite[Corollary 3.11]{Rains05} \[ \bar{P}_{\mu}^{\ast}\big(z(1,t,\dots,t^{n-1});q,t,s\big) =\frac{t^{2n(\mu)} q^{-n(\mu')}}{(-st^{n-1})^{\abs{\mu}}}\cdot \frac{(t^n,s/z,szt^{n-1};q,t)_{\mu}} {C_{\mu}^{-}(t;q,t)} \]

history

Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system $BC_n$
The various families of Macdonald (orthogonal) polynomials for classical root systems are all contained in the Koornwinder polynomials, and for a long time it was assumed they represented the highest possible level of generalisation.

related items

encyclopedia

https://en.wikipedia.org/wiki/Koornwinder_polynomials

expositions

Rains, Eric M. "Elliptic Analogues of the Macdonald and Koornwinder Polynomials." Proceedings of the International Congress of Mathematicians. Vol. 901. No. 2010. 2010. http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.2530.2554.pdf
Stokman, Jasper V. “Lecture Notes on Koornwinder Polynomials.” In Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207. Adv. Theory Spec. Funct. Orthogonal Polynomials. Nova Sci. Publ., Hauppauge, NY, 2004. http://www.ams.org/mathscinet-getitem?mr=2085855.
Stokman, Jasper V. “Macdonald-Koornwinder Polynomials.” arXiv:1111.6112 [math], November 25, 2011. http://arxiv.org/abs/1111.6112.

articles

van Diejen, J. F., and E. Emsiz. “Branching Rules for Symmetric Hypergeometric Polynomials.” arXiv:1601.06186 [math-Ph], January 22, 2016. http://arxiv.org/abs/1601.06186.
Corteel, Sylvie, and Lauren Williams. ‘Macdonald-Koornwinder Moments and the Two-Species Exclusion Process’. arXiv:1505.00843 [cond-Mat, Physics:nlin], 4 May 2015. http://arxiv.org/abs/1505.00843.
Stokman, Jasper, and Bart Vlaar. “Koornwinder Polynomials and the XXZ Spin Chain.” Journal of Approximation Theory 197 (September 2015): 69–100. doi:10.1016/j.jat.2014.03.003.
van Diejen, J. F., and E. Emsiz. “Branching Formula for Macdonald-Koornwinder Polynomials.” arXiv:1408.2280 [math], August 10, 2014. http://arxiv.org/abs/1408.2280.
Rains, Eric M. “BCn-Symmetric Polynomials.” Transformation Groups 10, no. 1 (March 2005): 63–132. doi:10.1007/s00031-005-1003-y. http://arxiv.org/abs/math/0112035.
[Mimachi01] K. Mimachi, A duality of Macdonald--Koornwinder polynomials and its application to integral representations, Duke Math. J.107 (2001), 265--281.
Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267--282.
[Okounkov98] * Okounkov, A. “BC-Type Interpolation Macdonald Polynomials and Binomial Formula for Koornwinder Polynomials.” Transformation Groups 3, no. 2 (June 1998): 181–207. doi:10.1007/BF01236432.
J. F. van Diejen, Self-dual Koornwinder--Macdonald polynomials, Invent. Math. 126 (1996), 319--339.
van Diejen, J. F. “Commuting Difference Operators with Polynomial Eigenfunctions.” arXiv:funct-an/9306002, June 7, 1993. http://arxiv.org/abs/funct-an/9306002.
[Koornwinder92] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, pp. 189–204, Contemp. Math. 138, Amer. Math. Soc., Providence, 1992. http://oai.cwi.nl/oai/asset/2292/2292A.pdf
[Gustafson90] R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97--105.